424 Dr BromwicJi, Symbolical methods 



to the symbolical manipulation oif(q). For, on tliis path we can 

 '^^^^^ V = -\- id (on the upper straight part), 



or — id (on the lower straight part) 



and X= — kO-; 



thus the convergence is always ensured by the presence of the 

 exponential e-^^'K 



With these general remarks we proceed now to evaluate the 

 special functions used. 



(i) Interpretation of q. 

 This becomes 



1 r ,,<^A 1 [\ „. .^,2de , If",,., ,,.2dd 



2 [-^_^,,. ^, 2 1 V- 1 . 



TTJo 7T^{Kt) 2 ^{TTKty 



the contribution from the small circle there tends to zero with 

 the radius (its value is in fact proportional to the square-root of 

 the radius). 



Hence q = l/\/{7TKt). 



(ii) Interpretation of q^. 



Repeating the foregoing transformations, we obtain 



" ' d'^e-'i^'dd 



rr J Kdt ■\/{7TKt) ' 

 and so q^ = 



2^7T\Kt)^' 



(iii) Interpretation of gSw+i^ 

 Similarly 



o2^+i = 1 (IT^±_ ^ , . .^ 1 . 3 ... (2m - 1) 



It should be noticed that symbohcally the relation 



- {'- 



is obvious, since Kq^ = p. 



(iv) Interpretation of q^, q^, .... 



It is here easy to verify that the two integrals from the upper 

 and lower paths cancel; so that q^ -- 0, g* = 0, etc. This again is 



obvious symbolically, since q^ -^ - ^ (l) ^ 



K ot 



?'"^' = .Tr«l.-J ? 



(v) Interpretation of 1/q. 



This is ±r-''7-v,..^ 



